Re: Is a function a relation?
Date: Tue, 23 Jun 2009 04:17:14 -0700 (PDT)
On 23 juin, 12:33, David BL <davi..._at_iinet.net.au> wrote:
> On Jun 23, 4:34 pm, Cimode <cim..._at_hotmail.com> wrote:
> > On 23 juin, 08:14, David BL <davi..._at_iinet.net.au> wrote:> On Jun 23, 1:35 pm, David BL <davi..._at_iinet.net.au> wrote:
> > > > Yes that's one way of looking at it.
> > > I'll expand on what I mean by that. It seems to me that one could use
> > > special conventions to "show" that just about any type can be regarded
> > > as a specialisation of a relation. E.g. one could say that a whole
> > > number in [0,255] is a relation by introducing symbols to represent
> > > 1,2,4,8,...,128 and the relation records a set of symbols that are
> > > then interpreted in the manner of an 8 bit unsigned representation.
> > Relations is a possible construct that can represent *any* type if we
> > are to consider that a type is a set of values. Nevertherless, a
> > logical computing model (to define among other things the physical
> > reprentation of domain values) must be defined first (that is what I
> > spent the last 10 years working onto)...Hope this helps...
> It could be thought that the logical can only exist as an abstraction
> over the physical. However I don't believe that's a useful way to
> think. In fact I suggest it misses the idea behind physical
> independence. What I mean is that the logical doesn't need to be
> "realised" or "reified" by the physical at all!
> This could be seen as just a metaphysical comment (more specifically
> in favour of mathematical realism), but what I really mean is that
> pure mathematical systems can for example define things like the
> integers in a way that's unique up to isomorphism through the
> axiomatic approach, and that perspective is all one needs at the
> logical level. I don't see how the physical comes into it at all.
> Putting it another way (using the language of a mathematical realist
> in denial), database values don't exist in time and space!
> It's not clear that type systems particularly help in this purist
> mathematical endeavour.
The problem of achieving physical/logical independence under the assumption of RMis far more complex than a simple taxonomy exercice. The need for a specialized computing model is a prerequisite for implementing relational logical operations.
The computing model must both take into considerations physical limitations of current systems while allowing sound representation of logical relational operations.
> I note that the usual axioms of set theory
> completely ignore any concept of type. I'd be interested to know
> whether modern mathematicians that have researched type theory believe
> it's important to mathematical foundations. My understanding is that
> Russell only investigated type theory with the aim to avoid paradoxes
> by preventing loops, but his work was made redundant by axiomatic
> systems like ZFC which is believed to be free of paradoxes.
RM is a part of set theory. Domains are in fact types that are defined in a naive way. Received on Tue Jun 23 2009 - 13:17:14 CEST